Method to estimate the rise-time of a pulse for single and multi-channel data

ABSTRACT

Techniques, systems, architectures, and methods for estimating the rise time of a pulse for multi-channel and signal channel cases involving an optimization system and method that can be solved iteratively based on sparse priors for a wide span of signal-to-noise ratios.

FIELD OF THE DISCLOSURE

The following disclosure relates generally to radar systems and, morespecifically, to systems and method to estimate the rise-time of a pulsefor single and multi-channel data.

BACKGROUND

Accurate estimation of the rise-time of a pulse when the received pulsecharacteristic is not known can be useful in different applications. Forexample, current, state of the art radar systems have the capability torapidly change their center frequency and modulation type. Consequently,systems and methods of associating the received pulses with differentradar types is fundamentally important to identifying and geolocatingthem and could be used alongside traditional feature identifiers such ascenter frequency, Pulse Repetition Interval (PRI), pulse length,modulation type, etc. For example, for radar systems, where therise-time of a pulse is impervious to frequency agility, PRI, andpulse-width, the rise-time of a pulse (also herein referred to simply asrise-time) can provide an independent feature, thereby increasingidentification accuracy.

Current systems, however, do not use rise-time as part of the PulseDescriptor Word (PDW), a description of the properties of a signal thatmay include amplitude, frequency, direction, etc. and that may be usedto identify an unknown signal. This is primarily due to estimationerrors in noise and interference.

What is needed, therefore, are systems and methods to accuratelyestimate the rise-time of a pulse for single and multi-channel data inboth low and high signal-to-noise ratio (SNR) regimes while minimizingestimation errors due to noise and interference.

SUMMARY

An optimization system and method that can be solved iteratively basedon sparse priors to estimate the rise-time of a pulse for a wide span ofsignal-to-noise ratios is herein disclosed. The method can be used asboth a multi-channel and signal channel case, depending on accuracy vscomputational complexity. Experimental results with a pulse of arise-time of 50 nanoseconds are then provided.

Furthermore, while discussed primarily in the context of identificationof unknown radar signatures, such techniques could also be used inwireless communications to differentiate different signals coming fromdifferent devices and in other applications where distinguishing betweendifferent signals is desired.

The features and advantages described herein are not all-inclusive and,in particular, many additional features and advantages will be apparentto one of ordinary skill in the art in view of the drawings,specification, and claims. Moreover, it should be noted that thelanguage used in the specification has been selected principally forreadability and instructional purposes and not to limit the scope of theinventive subject matter.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flowchart describing a method of JGSDD in accordance withembodiments of the present disclosure;

FIG. 2 is illustrates the pulse on two channels with a relative delaybased on the parameters tabulated in Table 1;

FIG. 2 is a plot describing ideal, noisy, and estimated signals with thecharacteristics described in Table 1 and FIG. 2;

FIG. 3 is a plot showing a Monte Carlo simulation describing meanrise-time of the signal of FIG. 3 with the characteristics described inTable 1 and FIG. 2 over 200 Simulation points, as a function ofSignal-to-Noise Ratio (SNR); and

FIG. 4 is a plot showing the root-mean square (RMS) error of therise-time estimator of FIG. 3 from the truth over 200 simulation pointsas a function of SNR; and

FIG. 5 is a plot illustrating the root-mean-square error of both singleand dual channel rise-time estimators.

These and other features of the present embodiments will be understoodbetter by reading the following detailed description, taken togetherwith the figures herein described. The accompanying drawings are notintended to be drawn to scale. For purposes of clarity, not everycomponent may be labeled in every drawing.

DETAILED DESCRIPTION

A variety of acronyms are used herein to describe both the subject ofthe present disclosure and background therefore. A brief listing of suchacronyms is provided below:

-   -   ADMM—Alternating Direction Method of Multipliers;    -   JGSDD—Joint Group-Sparse Denoising and Delay, a method of        determining pulse rise-time based on iterative, convex        optimization techniques;    -   NC—Non-Convex;    -   NC-GTVD—A more generalized form of TVD where the penalty        function is non-separable and non-convex;    -   PDW—Pulse Descriptor Word;    -   PRI—Pulse Repetition Interval;    -   Rise-time—The time it takes for a pulse to reach from 10% to 90%        maximum height. It is estimated after the denoising algorithm        presented in this patent; these reference levels can be changed        to 20% to 80% or any arbitrary reference level of the rise-time        of the pulse.    -   RMS—Root-Mean Square;    -   SNR—Signal to Noise Ratio;    -   TDOA—Time Difference of Arrival; and    -   TVD—Total Variation Denoising—A Signal Processing optimization        technique that penalizes the variation of the signal in        combination with its first derivative coefficients assumed to be        sparse in order to estimate it.

The rise-time of the pulse can be used as an additional discriminatorbetween different radar types. This is especially useful as therise-time is independent of the center frequency of the pulse and thusfrequency agility and other features that are used by some radars toconfuse the EW receivers are not effective where such a discriminator isused.

More specifically, we disclose systems and methods involving the use ofan algorithm to determine the rise time of a pulse where there are atleast two noisy signals y₁ and y₂ measured across different antennaelements and it is assumed that that one channel is a delayed amplitude,scaled version of the other channel. We use the term S^(−α) to denote adelay operator where S^(−α)y₂=ζy₂(n−α) and assume that when there are noadditive noise present, y₁=ζy₂(n−α) for some (potentially complex)amplitude scaling factor and real value a. The variable a depends on theposition of the target and measuring platform, the base-line, channelpropagation conditions, sampling rate, waveform, frequency and otherparameters related to the over-all measurement geometry. The variable nrepresents the sample time. In other words we assume the model,y₁(n)=x(n)+v₁(n) and y₂(n)=ζ(S^(−α)x(n))+v₂(n), where v represents noiseand interference and x represents the original transmit signal.

We consider the optimization formulation and estimate x and a from thejoint data:

$\begin{matrix}{{F_{\lambda}\left\{ {y_{1},y_{2}} \right\}} = {\underset{x,\alpha,K}{argmin}\left\{ {{\frac{1}{2}{w_{1} \cdot {{y_{1} - x}}_{2}^{2}}} + {\frac{1}{2}{w_{2} \cdot {{y_{2} - x}}_{2}^{2}}} + {\lambda \cdot \left( {\varphi\left( {{{Dx};a},K} \right)} \right)}} \right\}}} & {{Equation}\mspace{14mu}(1)}\end{matrix}$

Here the penalty function (the regularization function), φ, can bechosen very arbitrarily, in embodiments depending on generalcharacteristics of the data, such as the data being represented by acoefficients in a filter or transform domain. As an example, thederivative of an ideal pulse is primarily zero, except for thetime-sample interval consisting of its fall-time and rise-time.Consequently, it makes sense to model a base-band pulse to be sparse(having mostly zero or near zero values) in the difference filterdomain. Or, for example, if a signal is sparse in the Fourier domain,the regularization function can be suitably chosen to model sparsity inthe frequency domain. Now regarding Equation 1, specifically, w₁, w₂,and A are weights for the data-fidelity terms and the regularizationfunction, respectively. They may be chosen as scalars or higherdimensional vectors, when appropriate. As special cases, note in theabove formulation, when one of w₁ or w₂ is set to zero, Equation (1)reduces to the single channel estimation case. Also, if theregularization parameter is set to zero, then least square estimation isperformed. The variables a, and K relate to the parameters of thepenalty function, as further described below, while is a complex scalingfactor. Extension of the optimization above to multiple channel datawould be within the skill of one of ordinary skill in the art.

As a non-limiting example, we choose an appropriate penalty function tobe such that it is assumed x represents data whose first orderdifference is sparse relative to its length with non-zero coefficientsclustered together. This is a general assumption, since most pulseradars have an off and on period with a fall and rise-time. Embodimentsalso take into account combinations of sparsity in the first or/andhigher order derivatives. For example, FIG. 2 shows a 50 nanosecondrise-time pulse (see Table 1 for parameters) for two different channels,and FIG. 3 shows the derivative of the pulse. As illustrated in FIG. 3,the pulse contains non-zero coefficients during the rise-time period andalso when the power on transmission switch is turned off to the radarlistening mode. FIG. 3 also illustrates that, during the rise-time, thefirst order difference of the pulse has multiple non-zero coefficientsthat are clustered together. We have chosen ζ=1 for the currentexperiment. Although it is possible to normalize the power of bothchannels to solve Equation (1), in some situations, it is advantageousto have the variable as part of the total optimization framework. Inaddition, embodiments utilize penalty functions that are non-convex.

In embodiments, non-convex penalty function with parameters such thatEquation (3) is still maintained as convex due to the combination of thedata-fidelity terms and other convex regularizers are used to avoidlocal-minima. For example, the following non-convex penalty functionswith over-lapping structure sparsity are used in embodiments, althoughmany other cost functions would also be suitable, such as thosedescribed in the following texts, which are herein incorporated byreference, in their entirety:

-   -   a. Selesnick, Ivan W., and Ilker Bayram. “Sparse signal        estimation by maximally sparse convex optimization.” IEEE        Transactions on Signal Processing 62.5 (2014): 1078-1092; and    -   b. E. J. Candes, M. B. Wakin, and S. P. Boyd, “Enhancing        sparsity by reweighted L1 minimization,” J. Fourier Anal. Appl.,        vol. 14, no. 5-6, pp. 877-905, December 2008.

$\begin{matrix}{{\varphi\left( {x;a} \right)} = {\sum\limits_{i = 1}^{N}{\frac{2}{a\sqrt{3}}\left( {\left( {\arctan\frac{1 + {2a{f\left( {x_{i};K} \right)}}}{\sqrt{3}}} \right) - \frac{\pi}{6}} \right)}}} & {{Equation}\mspace{14mu}(2)}\end{matrix}$

Where f(x_(i);K) is chosen to promote over-lapping structure sparsityand is defined as:

$\begin{matrix}{{f\left( {x_{i};K} \right)} = \left\lbrack {\sum\limits_{k = 0}^{K - 1}{{x\left( {i + k} \right)}}^{2}} \right\rbrack^{\frac{1}{2}}} & {{Equation}\mspace{14mu}(3)}\end{matrix}$

Here f represents a sliding Euclidean norm of size K that starts atsample i up to sample K+i−1 of the estimated sampled pulse x. Thenotation x_(i) is used as a convention to note that the samples start ati for the estimated pulse x.

Here, f(x;K) can still be more generalized by using hamming (or other)types of weighting functions to weight the sum. Also, for K>1, we areassuming that the signal, x, has a cluster of close values. Lastly, thevariable “a” promotes non-convexity allowing for more robust estimationof the signal in noise.

An alternative function is:

$\begin{matrix}{{\varphi\left( {x;a} \right)} = {\sum\limits_{i = 1}^{N}\left\lbrack {\sum\limits_{k = 0}^{K - 1}{V_{k}{g\left( {{x\left( {i + k} \right)};a} \right)}^{p}}} \right\rbrack^{r}}} & {{Equation}\mspace{14mu}(4)}\end{matrix}$

Where V_(k) is a window (e.g. rectangular, hamming, etc.) and g(x) iscan be chosen as a convex or non-convex function, and p and r are realnumbers. As an example, g(x;a) can be chosen as although not limitedtoo:

$\begin{matrix}{{g\left( {x;a} \right)} = \frac{1}{{x}\left( {1 + {a{x}} + {a^{2}{x}^{2}}} \right)}} & {{Equation}\mspace{14mu}(5)}\end{matrix}$

Selection of the regularization function (φ(Dx; a, K) in Equation (1))will affect performance because different regularization functions modeldifferent types of apriori knowledge about the pulse. For the exampleresults described herein, we first define a mixed norm in Equation (6)for our penalty function.

$\begin{matrix}{{\varphi_{G}(x)} = {\sum\limits_{i = 1}^{N}\;\left\lbrack {\sum\limits_{k = 0}^{K - 1}\;{V_{k}{{x\left( {i + k} \right)}}^{2}}} \right\rbrack^{1/2}}} & {{Equation}\mspace{14mu}(6)}\end{matrix}$

Rather than penalizing high values of φ_(G)(x) in Equation (1), we maypenalize φ_(G)(D^(m)x) where D^(m) is the mth order difference operator,which can be written as a difference matrix. For m=1, D is the (N−1)×Ndifference matrix where the first row is [−1 1 zeros(1,N−2)]. D² issimilarly defined as a (N−2)×N matrix with the first row [−1 2−1zeros(1,N−3)]. Each additional row is then shifted by one zero to theright, relative to the previous row. The rows of the difference matrix,D, can be written as any filter, including a notch filter.

This model takes into account that the derivative of the pulse hasgroups of coefficients that are clustered together and is designed topenalize large clusters of non-zero derivative coefficients. For theexperimental results described herein, the first order differencepenalty function φ_(G)(Dx), which penalizes a cluster of derivativecoefficients is used in Equation (1) is used.

Now regarding Joint Group-Sparse Denoising and Delay (JGSDD) techniques,an iterative solution of Equation (1) consists of fixing a, solvingEquation (1) for multiple values of a, and choosing a vector, x, suchthat the cost function of Equation (1) is minimized. Such a solution isderived, in embodiments, by various methods including: MajorizationMinimization, Proximal Methods, Forward-Backward Splitting, and othernon-linear, convex optimization techniques. An algorithm usingMajorization-Minimization (denoted by joint group-sparse denoising anddelay (JGSDD) estimator representing an embodiment of the presentinvention is presented below to measure the rise-time from the output ofmultiple channels (Figueiredo, Mario AT, et al. “On total variationdenoising: A new majorization-minimization algorithm and an experimentalcomparison with wavelet denoising.” 2006 International Conference onImage Processing. IEEE, 2006.). Each estimate of x at each iteration L,is denoted by x^((L)). Here we assume that the SNR for both channels aresimilar; if they are estimated to be different, then the coefficientvectors of w₁ and w₂ can be set to a function of the SNR that is linearor quadratic or another arbitrary function designed to emphasize thehigher SNR channel relative to the lower SNR channel.

Now referring to FIG. 1, a method of JGSDD is described. Morespecifically, the method comprises:

Step AA) Obtain: y₁, y₂, where y₁ and y₂ are noisy, base-band samples ofa pulse;

Step BB) Initiliaze the values of K, λ, V, P, where K initializes agroup over-lap parameter, Δ is a regularization parameter, V is aweighting vector that represents uniform, hamming, hanning, or othertypes of weighting parameter used in the regularization function ofEquation (6), P represents a set of P delay values such that T={τ₁, τ₂ .. . τ_(p)} and to initialize the value of a variable means to set thevector or scalar to a specific value;

Step CC) Outer loop: For each feasible value of τ_(p) indexed by p from1 to P (where p represents an index for the delay values);

Step A) Shift y₂ by τ_(p) sample delays, which, mathematically, can beexpressed as ý₂=y₂(n−τ_(p)) and where shifting a variable by anothervariable means to shift the samples of a signal by moving the signal tothe left or right along a discrete time axis;

Step B) Initialize the value of L=0 (where 1 represents the currentiteration of the loop);

Step C) Set x_(p) ⁽⁰⁾=0.5*(y₁+{grave over (y)}₂), where x_(p) ^((L))represents the estimate of x at the Lth iteration); Here we haveaveraged the shifted value y₂ derived from step A with that of y₁. Thisis done so that the best candidate estimate of x can be estimated fromthe two independent measurements.

Step D) Set y=x_(p) ⁽⁰⁾ (Initialization Parameter);

Step E) Set h=D^(T)y (Initialization Parameter)

Step F) Inner Loop: Repeat until convergence for L=1: N_(loop)

Step 1) Set u=D x_(p) ^((L−1)), where D is the first order differenceoperation and u is a temporary variable used in the optimization);

Step 2)

${\lbrack M\rbrack_{n,n} = {\sum\limits_{j = 0}^{K - 1}\;\left\lbrack {\sum\limits_{k = 0}^{K - 1}\;{{V_{k}{u\left( {n - j + k} \right)}}}^{2}} \right\rbrack^{- {.5}}}},$where u is weighted by a window V_(k), in embodiments rectangular,hamming, hanning, etc.;

Step 3)

${B = {{\frac{1}{\lambda}M^{- 1}} + {DD^{T}}}},$where B is a constant Matrix;

Step 4) x_(p) ⁽¹⁾=y−D^(T)(B⁻¹h);

Finish Inner Loop Step F:

Step DD) Evaluate

$\begin{matrix}{c_{p} = \left\{ {{\frac{1}{2}{{y_{1} - x}}_{2}^{2}} + {\frac{1}{2}{{{\overset{`}{y}}_{2} - x}}_{2}^{2}} + {\lambda\left( {\varphi_{G}\left( {Dx_{p}} \right)} \right)}} \right\}} & \;\end{matrix}$

Finish Outerloop from step CC:

Final Output is then provided in Step EE, below:

Step EE)

$x_{p^{*}} = \underset{p}{{\arg\;\min\; c_{p}},}$from the set of c_(p) where p represents different delay values τ_(p),choose the index p* that corresponds to the minimum value of c_(p). Thenx_(p*) is the optimum solution obtained from the algorithm to estimate xwhich is the original pulse.

It should be noted that, for the integer value N_(loop), one can choosea tolerance value that evaluates the successive difference of thesolution obtained in the current inner loop x_(p) ^((L)) relative to theprevious inner loop x_(p) ^((L−1)) for a norm such as the L2 norm.Alternatively, one can choose a number of iterations N_(loop) based torun the iterative optimization solution a priori. Yet another method isto evaluate the difference of c_(p) as function of successive values ofp and p−1 for a norm, such as the L2 norm. Other methods to evaluateconvergence to a unique solution would also be known to thoseknowledgeable of the arts.

Once x_(p*) is estimated, then we may estimate the rise-time of x_(p*)between two reference levels, such as the time from the pulses to risefrom 0.1 to the 0.9 amplitude levels (i.e. the rise time of the pulse).

Now regarding the special case of single channel estimation andrise-time estimation, using Equation (1), by setting w₂=0, one canestimate x₁. Similarly, by setting w₁=0, one can get an estimate of x₂and then use this estimate to estimate rise-time. Although variousmethods and definitions exist for rise-time and its measurement, toestimate the rise-time, here we provide a simple definition of measuringthe rise-time between the 0.1 and 0.9 amplitude levels of the peak.Other common reference levels such as 0.2 to 0.8 or 0.3 to 0.07amplitude levels can also be chosen in parallel for better estimationand feature classification of pulses, as needed.

In general, this method utilizes both data-sets independently and,theoretically, should not perform as well as joint-decoding. Thismethod, however, has the advantage of not requiring an optimization tobe solved for each a and provides results that are still significantlybetter than traditional cross-correlation techniques. The computationalcost is also reduced significantly by the number of candidate delaysthat are used to solve Equation (1).

One method to speed up the joint estimation process of embodiments is tosolve the single channel to obtain a narrower range of candidate delayvalues. Another method is to use parallelization, since each candidatedelay lends itself to a different optimization program.

Now regarding obtaining the solution of the optimization, a variety ofapproaches including ADMM, Proximal Splitting Methods, MajorizationMinimization, brute-force search, and others, as would be known to oneof ordinary skill in the art, can be used to solve Equation (1). Theseand other approaches can be found described in the following texts,which are herein incorporated by reference, in their entirety: Afonso,Manya V., Jośe M. Bioucas-Dias, and Mario AT Figueiredo. “An augmentedLagrangian approach to the constrained optimization formulation ofimaging inverse problems.” IEEE Transactions on Image Processing 20.3(2011): 681-695; Combettes, Patrick L., and Jean-Christophe Pesquet.“Proximal splitting methods in signal processing.” Fixed-pointalgorithms for inverse problems in science and engineering. Springer,New York, N.Y., 2011. 185-212; and Figueiredo, Mario AT, Jose M.Bioucas-Dias, and Robert D. Nowak. “Majorization—minimization algorithmsfor wavelet-based image restoration.” IEEE Transactions on Imageprocessing 16.12 (2007): 2980-2991.

It should be noted that, given the filter operations are bandedmatrices, the solution of Equation (1) can, in general, involve bandedmatrixes, which are efficient from a numerical computationalimplementation.

Now regarding our sample experimental result, we illustrate theaforementioned methods and systems that make use of them by thefollowing rise-time experiment, which is tabulated in Table 1, below.

TABLE 1 Parameters of the rise-time experiment Signal-to- Sampling NoiseRatio Rise-time Delay Pulse-Width Rate (dB) 50 0.02 0.5994 1600 [−6:20]dB nanoseconds microseconds microseconds MHz

Now specifically referring to FIG. 2, FIG. 2 illustrates the pulse onboth channels with a relative delay based on the parameters tabulated inTable 1. FIG. 3 shows a noisy pulse, an estimated pulse, and an idealpulse using the single channel algorithm. FIG. 4 shows the estimatedmean rise-time using both the single channel and multi-channel (in thiscase two) estimator over 200 simulation points for each SNR. We notethat the mean rise time is correctly estimated for the dual channel caseat SNR of 0 dB. Consequently, we may use the algorithm over multiplepulses and take a mean of the individual rise-times to obtain betteraccuracy Finally, FIG. 5 illustrates the root-mean-square error of boththe single and dual channel rise-time estimators.

The foregoing description of the embodiments of the present disclosurehas been presented for the purposes of illustration and description. Itis not intended to be exhaustive or to limit the present disclosure tothe precise form disclosed. Many modifications and variations arepossible in light of this disclosure. It is intended that the scope ofthe present disclosure be limited not by this detailed description, butrather by the claims appended hereto.

A number of implementations have been described. Nevertheless, it willbe understood that various modifications may be made without departingfrom the scope of the disclosure. Although operations are depicted inthe drawings in a particular order, this should not be understood asrequiring that such operations be performed in the particular ordershown or in sequential order, or that all illustrated operations beperformed, to achieve desirable results.

What is claimed is:
 1. A method of estimating a rise-time of a pulse forsingle and multi-channel data, the method comprising: receiving a firstsignal, y₁, and a second signal, y₂, wherein the first and second signalare signals received on an antenna of a radar system; using thefollowing equation:${F_{\lambda}\left\{ {y_{1},y_{2}} \right\}} = {\underset{x,\alpha,K,C}{argmin}\left\{ {{\frac{1}{2}{w_{1} \cdot {{y_{1} - x}}_{2}^{2}}} + {\frac{1}{2}{w_{2} \cdot {{y_{2} - x}}_{2}^{2}}} + {\lambda \cdot \left( {\varphi\left( {{{Dx};a},K} \right)} \right)}} \right\}}$where w₁ and w₂ are weights for the data-fidelity terms, where λ is aweight for the regularization function, where C is an amplitude scalingfactor, where φ (x; a)represents a regularization function, where D is adifference matrix, where K initializes a group over-lap parameter, andwhere α is a real value; fixing α; solving for multiple values of α;choosing a vector, x, such that a cost function is minimized; andestimating the rise time of the pulse; discriminating, by the radarsystem, between a plurality of radar types based on the estimated risetime of the pulse; wherein the first signal and the second signal aremeasured across different antenna elements of the radar system; andwherein one of the first signal or second signal is a delayed amplitude,scaled version of the other signal.
 2. The method of claim 1, whereiny₁(n)=x(n)+v₁(n) and y₂(n)=ζ(S^(−α)x(n))+v₂ (n), where S^(−α) denotes adelay operator, v represents noise and interference, and x represents anoriginal version of the first signal and the second signal.
 3. Themethod of claim 1 wherein the regularization function is convex ornon-convex.
 4. The method of claim 1 wherein the regularization functionis chosen such that it is assumed x is a data whose first orderdifference is sparse relative to its length, and its non-zerocoefficients are clustered together.
 5. The method of claim 1 whereinsolving the equation comprises the use of a method chosen from the groupconsisting of Majorization, ADMM, Proximal Methods, and non-linear,iterative convex optimization techniques.
 6. The method of claim 1wherein the regularization function comprises:${\varphi\left( {x;a} \right)} = {\sum\limits_{i = 1}^{N}{\frac{2}{a\sqrt{3}}\left( {\left( {\arctan\frac{1 + {2a{f\left( {x_{i};K} \right)}}}{\sqrt{3}}} \right) - \frac{\pi}{6}} \right)}}$and wherein f(x_(i);K) is chosen to promote over-lapping structuresparsity and is defined as:${f\left( {x_{i};K} \right)} = {\left\lbrack {\sum\limits_{k = 0}^{K - 1}\left| {x\left( {i + k} \right)} \right|^{2}} \right\rbrack^{\frac{1}{2}}.}$7. The method of claim 6 wherein solving for f(x_(i);K) comprises theuse of an approach selected from the group consisting of AlternatingDirection Method of Multipliers, Proximal Splitting, Majorization,Minimization, and Brute-Force.
 8. The method of claim 7 whereinf(x_(i);K) is generalized through the use of a weighting function toweight the sum.
 9. The method of claim 8 wherein the weighting functionis a hamming-type function.
 10. The method of claim 1 wherein theregularization function comprises:${{\varphi\left( {x;a} \right)} = {\sum\limits_{i = 1}^{N}\left\lbrack {\sum\limits_{k = 0}^{K - 1}{V_{k}{g\left( {{x\left( {i + k} \right)};a} \right)}^{p}}} \right\rbrack^{r}}},$where: V_(i) is a window, p and r are real numbers, and g (x) is aconvex or non-convex function.
 11. The method of claim 10 wherein${g\left( {x;a} \right)} = {\frac{1}{{x}\left( {1 + {a{x}} + {a^{2}{x}^{2}}} \right)}.}$12. A method of estimating the rise-time of a pulse for single andmulti-channel data, the method comprising: receiving a first signal, y₁,and a second signal, y₂, wherein the first and second signal arereceived on an antenna coupled to a radar system; using the followingequation:${F_{\lambda}\left\{ {y_{1},y_{2}} \right\}} = {\underset{x,\alpha,K,C}{argmin}\;\left\{ {{\frac{1}{2}{w_{1} \cdot {{y_{1} - x}}_{2}^{2}}} + {\frac{1}{2}{w_{2} \cdot {{y_{2} - x}}_{2}^{2}}} + {\lambda \cdot \left( {\varphi\left( {{{Dx};a},K} \right)} \right)}} \right\}}$where w₁ and w₂ are weights for the data-fidelity terms, where λ is aweight for the regularization function, where C is an amplitude scalingfactor, where φ(x; a)represents a regularization function, where D is adifference matrix, where K initializes a group over-lap parameter, andwhere α is a real value; fixing α; solving for multiple values of α;choosing a vector, x, such that a cost function of is minimized; andestimating the rise time of the pulse; discriminating, by the radarsystem, between a plurality of radar types based on the estimated risetime of the pulse; wherein the first signal and the second signal aremeasured across different antenna elements of the radar system; whereinone of the first signal or second signal is a delayed amplitude, scaledversion of the other signal; wherein y₁(n)=x(n)+v₁(n) andy₂(n)=ζ(S^(−α)x(n))+v₂(n), where S^(−α) denotes a delay operator, vrepresents noise and interference, and x represents an original versionof the first signal and the second signal; wherein the regularizationfunction is chosen such that it is assumed x is a data whose first orderdifference is sparse relative to its length, and its non-zerocoefficients are clustered together; wherein the regularization functioncomprises:${{\varphi\left( {x;a} \right)} = {\sum\limits_{i = 1}^{N}{\frac{2}{a\sqrt{3}}\left( {\left( {\arctan\frac{1 + {2a{f\left( {x_{i};K} \right)}}}{\sqrt{3}}} \right) - \frac{\pi}{6}} \right)}}};$and wherein f(x_(i);K) is chosen to promote over-lapping structuresparsity and is defined as:${f\left( {x_{i};K} \right)} = {\left\lbrack {\sum\limits_{k = 0}^{K - 1}\left| {x\left( {i + k} \right)} \right|^{2}} \right\rbrack^{\frac{1}{2}}.}$13. A method of estimating the rise-time of a pulse for single andmulti-channel data, the method comprising: receiving a first signal, y₁,and a second signal, y₂, on an antenna of a radar system; using thefollowing equation:${F_{\lambda}\left\{ {y_{1},y_{2}} \right\}} = {\underset{x,\alpha,K,C}{argmin}\left\{ {{\frac{1}{2}{w_{1} \cdot {{y_{1} - x}}_{2}^{2}}} + {\frac{1}{2}{w_{2} \cdot {{y_{2} - x}}_{2}^{2}}} + {\lambda \cdot \left( {\varphi\left( {{{Dx};a},K} \right)} \right)}} \right\}}$where w₁ and w₂ are weights for the data-fidelity terms, where λ is aweight for the regularization function, where C is an amplitude scalingfactor, where φ(x; a)represents a regularization function, where D is adifference matrix, where K initializes a group over-lap parameter, andwhere α is a real value; fixing α; solving for multiple values of α;choosing a vector, x, such that a cost function of is minimized; andestimating the rise time of the pulse; wherein the first signal and thesecond signal are measured across different antenna elements of theradar system; wherein one of the first signal or second signal is adelayed amplitude, scaled version of the other signal; whereiny₁(n)=x(n)+v₁(n) and y₂(n)=ζ(S^(−α)x(n))+v₂(n), where S^(−α) denotes adelay operator, v represents noise and interference, and x represents anoriginal version of the first signal and the second signal; wherein theregularization function is chosen such that it is assumed x is a datawhose first order difference is sparse relative to its length, and itsnon-zero coefficients are clustered together; wherein the regularizationfunction comprises:${{\varphi\left( {x;a} \right)} = {\sum\limits_{i = 1}^{N}\left\lbrack {\sum\limits_{k = 0}^{K - 1}{V_{k}{g\left( {{x\left( {i + k} \right)};a} \right)}^{p}}} \right\rbrack^{r}}},$where: V_(i) is a window, p and r are real numbers, and g (x) is aconvex or non-convex function, and wherein${{g\left( {x;a} \right)} = \frac{1}{{x}\left( {1 + {a{x}} + {a^{2}{x}^{2}}} \right)}}.$14. The method of claim 1, performing a single channel estimation bysetting w₁ equal to zero and estimating a vector x₁, setting w₂ is setto equal zero, and estimating a vector x₂, and then estimating the risetime.
 15. The method of claim 1, wherein the estimated rise time of thepulse is used for geolocation of a source of the first signal and thesecond signal.
 16. The method of claim 1, wherein the estimated risetime of the pulse is used for identification of a source of the firstsignal and the second signal.
 17. The method of claim 1, wherein theestimating of the rise time of the pulse is calculated between 0.1 and0.9 amplitude levels of the peak.
 18. The method of claim 1, wherein theestimating of the rise time of the pulse is calculated between 0.2 and0.8 amplitude levels of the peak.
 19. The method of claim 1, wherein theestimating of the rise time of the pulse is calculated between 0.3 and0.7 amplitude levels of the peak.